Question

solve the equation.
$3^{x^2 - 24} = 9^x$
the solution set is {}.
(simplify your answer. use a comma to separate answers as needed.)

Explanation:

Step1: Rewrite bases as powers of 3

Since \(9 = 3^2\), we can rewrite \(9^x\) as \((3^2)^x\). Using the exponent rule \((a^m)^n = a^{mn}\), this becomes \(3^{2x}\). So the equation \(3^{x^2 - 24}=9^x\) is equivalent to \(3^{x^2 - 24}=3^{2x}\).

Step2: Set exponents equal (one - to - one property)

For exponential functions with the same base \(a>0,a
eq1\), if \(a^m = a^n\), then \(m = n\). Here, \(a = 3\), so we set the exponents equal: \(x^2-24 = 2x\).

Step3: Rearrange into quadratic equation

Subtract \(2x\) from both sides to get \(x^2-2x - 24=0\).

Step4: Factor the quadratic equation

We need two numbers that multiply to \(- 24\) and add to \(-2\). The numbers are \(-6\) and \(4\). So, \(x^2-2x - 24=(x - 6)(x + 4)=0\).

Step5: Solve for x

Set each factor equal to zero:

• If \(x - 6=0\), then \(x = 6\).
• If \(x + 4=0\), then \(x=-4\).

Answer:

\(6, - 4\)